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Write Minors and Cofactors of the elements of following determinants: (ii) determinant a c b d

Q : 1         Write Minors and Cofactors of the elements of following determinants:

               \small (ii)\begin{vmatrix} a &c \\ b &d \end{vmatrix}

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GIven determinant: \begin{vmatrix} a &c \\ b &d \end{vmatrix}

Minor of element a_{ij} is M_{ij}.

Therefore we have 

M_{11} = minor of element a_{11} = d

M_{12} = minor of element a_{12} = b

M_{21} = minor of element a_{21} = c

M_{22} = minor of element a_{22} = a

and finding cofactors of a_{ij} is  A_{ij} = (-1)^{i+j}M_{ij}.

Therefore we have:

A_{11} = (-1)^{1+1}M_{11} = (-1)^2(d) = d

 A_{12} = (-1)^{1+2}M_{12} = (-1)^3(b) = -b

A_{21} = (-1)^{2+1}M_{21} = (-1)^3(c) = -c

A_{22} = (-1)^{2+2}M_{22} = (-1)^4(a) = a

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